Finite Group Databasebeta

← #13048741

Subgroups — #13048741

Order 1296.

332 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C9×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×C4×C3⋊D9C211yesmaximal, characteristic, complemented
3C3×#6226C211yesmaximal, characteristic, complemented
4#10619899C212yesmaximal, complemented
5#10619898C212yesmaximal, complemented
6#89041—31nomaximal
7#88639—31nomaximal
8#88638—31nomaximal
9C2×C9×C3⋊D3C2212yes—
10C2×C3×C3⋊D9C2211yescharacteristic
11C32×C4×C9C2211yescharacteristic
12C3×C3⋊Dic9C2212yes—
13C3×C3⋊Dic9C2211yescharacteristic
14C3×C4⋊(C3⋊D3)D311yescharacteristic
15C3×C4×C3⋊D3—31no—
16C3×#634—31no—
17#6295—32no—
18#6294—32no—
19C4×C3⋊D9D311yescharacteristic, complemented
20#6226D311yescharacteristic, complemented
21C9×D12—31no—
22C9×D12—31no—
23C9×D12—61no—
24C3×C4×D9—31no—
25C3×C4×D9—61no—
26C3×Dic18—31no—
27C3×Dic18—61no—
28#6195—32no—
29#6194—32no—
30D3×Dic9—32no—
31D3×Dic9—32no—
32C2×C32×C9C2311yescharacteristic
33C9×C3⋊D3—22no—
34C3×C3⋊D9—21no—
35#3799—91no—
36#3796—91no—
37#3795—91no—
38#3699—91no—
39C2×C3×C3⋊D3—31no—
40C2×C3×C3⋊D3C2×D312yes—
41C33×C4C2×D311yescharacteristic
42C3×C3⋊Dic3—31no—
43C3×C3⋊Dic3—32no—
44C2×C3⋊D9C2×D311yescharacteristic
45C2×D3×C9—32no—
46C2×D3×C9—32no—
47C2×D3×C9—62no—
48C2×C3×D9—31no—
49C2×C3×D9—61no—
50C3×C4×C9—21no—
51C3×C4×C9—21no—
52C3×C4×C9—21no—
53C3×C4×C9—41no—
54C3×C4×C9C2×D311yescharacteristic
55C3×C4×C9C2×D311yescharacteristic
56C3⋊Dic9C2×D311yescharacteristic
57C3⋊Dic9C2×D312yes—
58C3×Dic9—31no—
59C3×Dic9—32no—
60C3×Dic9—62no—
61C3×Dic9—61no—
62C32×C9#4111yescharacteristic, complemented, Sylow
63C4⋊(C3⋊D3)D911yescharacteristic, complemented
64C4×C3⋊D3—31no—
65#634—31no—
66C3×D12—31no—
67C3×D12—31no—
68C3×D12—61no—
69C3×C4×D3—91no—
70C3×C4×D3—91no—
71C3×C4×D3—181no—
72C3×Dic6—91no—
73C3×Dic6—91no—
74C3×Dic6—181no—
75#626—92no—
76#626—92no—
77#625—92no—
78#624—92no—
79D3×Dic3—92no—
80D3×Dic3—92no—
81D4×C9—91no—
82#611—92no—
83C2×Dic9—92no—
84C4×D9—31no—
85C4×D9—61no—
86Dic18—31no—
87Dic18—61no—
88C2×C33C22×D311yescharacteristic
89C3×C3⋊D3—22no—
90C3×C3⋊D3—61no—
91C2×C3×C9—21no—
92C2×C3×C9—21no—
93C2×C3×C9—21no—
94C2×C3×C9—41no—
95C2×C3×C9C22×D311yescharacteristic
96C2×C3×C9C22×D311yescharacteristic
97C3⋊D9—21no—
98D3×C9—62no—
99D3×C9—62no—
100D3×C9—122no—
101C3×D9—62no—
102C3×D9—61no—
103#239—271no—
104#237—271no—
105#235—271no—
106C2×C3⋊D3—31no—
107C2×C3⋊D3C2×D912yes—
108C2×C3×D3—32no—
109C2×C3×D3—32no—
110C2×C3×D3—62no—
111C2×C3×D3—91no—
112C2×C3×D3—91no—
113C2×C3×D3—181no—
114C32×C4—21no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4—41no—
118C32×C4C2×D911yescharacteristic
119C32×C4D3211yescharacteristic
120C32×C4D3211yescharacteristic
121C3⋊Dic3—31no—
122C3⋊Dic3—32no—
123C3×Dic3—91no—
124C3×Dic3—92no—
125C3×Dic3—91no—
126C3×Dic3—92no—
127C3×Dic3—182no—
128C3×Dic3—181no—
129C22×C9—92no—
130C2×D9—31no—
131C2×D9—61no—
132C4×C9—21no—
133C4×C9—21no—
134C4×C9—41no—
135C4×C9D3211yescharacteristic
136Dic9—32no—
137Dic9—31no—
138Dic9—62no—
139Dic9—61no—
140C33#23711yescharacteristic
141C3×C9—21no—
142C3×C9—21no—
143C3×C9—21no—
144C3×C9—41no—
145C3×C9#23511yescharacteristic, complemented
146C3×C9#23911yescharacteristic, complemented
147C3×Q8—271no—
148C3×D4—91no—
149C2×C3×C4—271no—
150#67—272no—
151#67—272no—
152C2×Dic3—272no—
153D12—31no—
154D12—31no—
155D12—61no—
156D12—272no—
157C4×D3—91no—
158C4×D3—91no—
159C4×D3—181no—
160C4×D3—272no—
161C4×D3—272no—
162Dic6—91no—
163Dic6—91no—
164Dic6—181no—
165C2×C32—21no—
166C2×C32—21no—
167C2×C32—21no—
168C2×C32—41no—
169C2×C32C22×D911yescharacteristic
170C2×C32C2×D3211yescharacteristic
171C2×C32C2×D3211yescharacteristic
172C3⋊D3—22no—
173C3⋊D3—61no—
174C3×D3—62no—
175C3×D3—62no—
176C3×D3—122no—
177C3×D3—182no—
178C3×D3—181no—
179C3×D3—181no—
180C2×C9—21no—
181C2×C9—21no—
182C2×C9—41no—
183C2×C9—182no—
184C2×C9C2×D3211yescharacteristic
185D9—62no—
186D9—61no—
187#41—811noSylow
188C22×C3—92no—
189C22×C3—271no—
190C2×D3—32no—
191C2×D3—32no—
192C2×D3—62no—
193C2×D3—91no—
194C2×D3—91no—
195C2×D3—181no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—41no—
200C3×C4—271no—
201C3×C4—272no—
202C3×C4D3×D911yescharacteristic
203C3×C4D3×D911yescharacteristic
204C3×C4#111311yescharacteristic
205Dic3—92no—
206Dic3—91no—
207Dic3—92no—
208Dic3—91no—
209Dic3—182no—
210Dic3—181no—
211C32—21no—
212C32—21no—
213C32—21no—
214C32—41no—
215C32#369911yescharacteristic, complemented
216C32#379511yescharacteristic
217C32#379611yescharacteristic
218C9—21no—
219C9—21no—
220C9—41no—
221C9#379911yescharacteristic, complemented
222Q8—271no—
223D4—91no—
224D4—812no—
225C2×C4—271no—
226C2×C4—812no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3—21no—
230C2×C3—41no—
231C2×C3—182no—
232C2×C3—541no—
233C2×C3C2×D3×D911yescharacteristic
234C2×C3C2×D3×D911yescharacteristic
235C2×C3C2×#111311yescharacteristic
236D3—62no—
237D3—62no—
238D3—122no—
239D3—182no—
240D3—181no—
241D3—181no—
242C22—92no—
243C22—271no—
244C4—271no—
245C4—272no—
246C4#6643111yescharacteristic
247C3—21no—
248C3—21no—
249C3—21no—
250C3—41no—
251C3#8863811yescharacteristic, complemented
252C3#8863911yescharacteristic, complemented
253C3#8904111yescharacteristic
254C2—182no—
255C2—541no—
256C2C2×#6643111yescharacteristic
subgroupG/H
centerC2C2×#66431
commutator, cosocleC2×C32×C9C23
FrattiniC2×C3C2×#1113
FittingC32×C4×C9C22
socleC2×C33C22×D3
hypercenterC4#66431
hypocenterC32×C9#41
LCSG ▷ C2×C32×C9 ▷ C32×C9
UCSC1 ◁ C2 ◁ C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C9×C4⋊(C3⋊D3)C2semidirect
2. C3×C4×C3⋊D9C2semidirect
3. C3×#6226C2semidirect
4. #10619899C2semidirect
5. #10619898C2semidirect
19. C4×C3⋊D9D3semidirect
20. #6226D3semidirect
62. C32×C9#41semidirect
63. C4⋊(C3⋊D3)D9semidirect
145. C3×C9#235semidirect
146. C3×C9#239semidirect
215. C32#3699semidirect
221. C9#3799semidirect
251. C3#88638semidirect
252. C3#88639semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C9×C4⋊(C3⋊D3)2
C3×C4×C3⋊D92
C3×#62262
#106198992
#106198982
#890413
#886393
#886383